The 20-year-old winger has been heavily linked with a move away from the Santiago Bernabeu having struggled to cement his place in Jose Mourinho's first-team this year, after initially making the breakthrough in 2011/12.
Gunners boss Arsene Wenger has been credited with an interest in bringing the Spain youth international to the Emirates Stadium, while Premier League rivals West Ham are also believed to be monitoring the youngster's situation.
However, both Arsenal and the Hammers will face a serious fight to secure Rodriguez's signature, with reports in Spain suggesting that Bayer Leverkusen are keen to sign him on a season-long loan deal.
The Bundesliga club are said to have made the Real Madrid starlet a priority target this summer, and hope the fact they already have one of Rodriguez's old youth team pals - Daniel Carvajal - in their ranks at the BayArena, will be enough to convince the player to make the temporary move to Germany.
It's thought that Real Madrid would prefer Rodriguez to leave the club on loan, due to the fact he is still regarded as a player with a big future in the game, meaning he could return to the Spanish capital in 12 months time with a year's experience of regular football.
That puts Bayer Leverkusen in the driving seat in what looks like being a three-way transfer tussle with Arsenal and West Ham for the former Castilla youth team star.